The figure illustrates a Keplerian orbit, with Cartesian coordinates (x,y) and p
ID: 2102747 • Letter: T
Question
The figure illustrates a Keplerian orbit, with Cartesian coordinates (x,y) andplane polar coordinates (r,phi)
.F = -(G*M*m)/r^2
The parametric equations for the orbit:
r(psi)=a(1-ecos(psi)) tan(phi/2)=[(1+e)/(1-e)]^1/2 * tan(psi/2) t(psi)=(T/(2pi))(psi-esin(psi)) where psi is the independent, parametric variable; a, e, and T are constants.
(A) What kind of curve is the orbit? Sketch a picture of the orbit that shows the meaning
of a and e. Explain the picture.
(B) Prove, directly from the parametric equations, that the angular momentum L is
constant. Express L in terms of the constants of the orbit (and any other relevant parameters).
(C) The energy E must also be constant. Derive, directly from the parametric equations,
an expression for E as a function of r.
(D) From the result of (C), and the fact that E must be constant, i.e., independent of r,
determine equations for E and T in terms of the constants of the orbit (and any other
relevant parameters).
For the next two equations, express the answers in terms of T and e.
(E) Determine the time for the planet to travel from perihelion (P) to aphelion (A).
(F) Determine the time for the planet to travel from C1 to C2.
I can't post the figure so C1 and C2 are North and south of Orbit respectivily. And A and P are West and East respectivily. If you only want to answer a few at least do Part (B) and (C) and I will give 5 stars.
Explanation / Answer
THIS WILL HELP YOU
2. Relevant equations
L = r x p
?net = r x F
3. The attempt at a solution
I know that in order for L to be constant, its derivative (net torque) must be 0, but this isn't really getting me anywhere. I know torque = rxF and L = rmv, but quite frankly I can't seem to figure out how to do this using the parametric equations.
What I have done using the parametric equations is this:
L = m*r(?)*v(t) = m*(a*(1-e*cos(?)))*v(t)
From here I'm having a difficult time finding an expression for v(t). I tried using the relations
?=((2*?)/T) and v = ?*r for the following work:
t(?) = (1/?)*(?-esin(?))
==> ? = (?-esin(?))/t(?)
==> ? = (?-esin(?))/((T/(2?))*(?-esin(?))
==> ? = (2?/T) (this obviously backtracked)
==> v = r*(2?/T)
?L = m*(a*(1-e*cos(?)))*(a*(1-e*cos(?)))*(2?/T)
or L = m*a2*(1-e*cos(?))2*(2?/T)